From Quantum Mechanics to String Theory Relativity (why it makes sense) Quantum mechanics: measurements and uncertainty Smashing things together: from Rutherford to the LHC Particle Interactions Quarks and the Strong Force Symmetry and Unification String Theory: a different kind of unification Extra Dimensions Strings and the Strong Force
The Higgs Mechanism Summary The weak force is different from the strong and EM forces because the mediators are massive, and carry charge particles are disturbances in fields; the elasticity in the field means a disturbance carries potential energy that we see as mass a spontaneously broken symmetry is a symmetry of the laws of nature that isn t realized in the world around us the Higgs field has a double well potential energy and when it settles into a minimum it breaks the symmetry between electromagnetic and weak forces because everywhere in the vacuum the Higgs field has a constant value, particles that interact with the Higgs get effective masses from the interaction energy
String Theory: A different kind of unification
String Theory: A different kind of unification The Trouble with Gravity
Gravity and Mass The mass of an object determines how much it is affected by gravity. An object of mass m in a gravitational field g feels a force of F = mg This is similar to electromagnetism, where the charge of an object q determines the force in an electric field E: F = qe However, the mass of an object m also determines its acceleration a when it feels a force F: F = ma This sets gravity apart: it means the acceleration of an object in a gravitational field is independent of the mass: F = ma = mg a = g
Equivalence Principle inertial mass (the m in F = ma) is the same as gravitational mass (the m in F = mg) Einstein s thought experiment: in empty space, a rocket accelerates forward with a person inside The person experiences this acceleration as a force towards the back of the rocket The person and all other objects in the rocket behave as if they are in a gravitational field: no experiment can distinguish between the effects of gravitation and acceleration gravitational effects can be absorbed into the background: the curvature of space (general relativity)
Newton s Constant Newton s constant G appears in the universal law of gravitation: F = Gm 1m 2 r 2 Quantum Mechanical decreasing size General Relativistic Quantum Gravity increasing space curvature It determines the strength of gravitational effects: when general relativity becomes important Classical increasing speed Special Relativistic Particle Physics Newton s constant G, the speed of light c, and Planck s constant h together create a framework of natural scales Gh Example: Planck length l p = = 10 35 meters c 3
Black Holes when a point mass curves spacetime, there is a singularity of curvature as you approach the mass usually, the singularity doesn t occur because the mass is spread out over some region of space (like a planet or a star) when the mass is concentrated in a small enough region of space, this is when a black hole occurs
Black Hole Horizon If a black hole has mass M then at a distance of R = GM/ c^2 there is a horizon objects, even light, can only pass inward across the horizon, not back out locally, particularly if the black hole is very big, the area around the horizon isn t that unusual. It s not highly curved and you wouldn t notice you had passed it for an observer outside the horizon, no experiment can reveal information about what occurs inside the horizon. Because of this we can t really ask questions about the singularity itself. This is called cosmic censorship
Hawking Radiation Classically, nothing can escape the horizon of a black hole. But quantum mechanically this is not quite true. a particle/anti-particle pair can be produced in the vacuum. Usually, this pair then annihilates again. But if the pair appears at the horizon of the black hole, one may fall inside the horizon and the other stay outside. In this case they cannot annihilate again. Until either of the virtual particles are observed, neither has definite energy. But after one falls into the black hole, the other is detected. At this point it has some positive energy. Conservation of energy then requires that the total energy (mass) behind the horizon decreases. By this process, the black hole radiates away mass.
Information Paradox In particle physics, every process is reversible: if we filmed it and reversed the film, the new process also makes sense. This is an assumption of quantum mechanics Black holes seem to violate this principle. According to GR, a black hole is completely described by a very few numbers (mass, charge, angular momentum, etc) Hawking radiation coming off the black hole is determined by only these numbers Complicated stuff falls in, simple stuff comes out: this is not reversible. Where does the information go?
Vacuum Energy Newton: gravitational force comes from mass Einstein: mass is a form of energy all energy gravitates Gravity is the one force that cares about total energy, not just energy differences. Suddenly we need to worry about the energy density of the vacuum. Higgs field energy Quantum fluctuations cause a background energy level All of these sources together mean that the vacuum should have some natural energy density (cosmological constant Λ) Positive energy (Λ < 0) or negative energy (Λ > 0)
The Cosmological Constant Λ effects the way the universe expands Ω matter normal matter makes the expansion decelerate (because of gravitation) Λ > 0 causes repulsive gravity. It can cause the universal expansion to accelerate measurement of the acceleration of expansion gives one relationship between normal matter and vacuum energy density matter and vacuum energy also causes the universe to curve. The measurement of this curvature gives a second relationship between normal matter and vacuum energy density measurements of dark and luminous matter gives a third constraint--together all imply positive cosmological constant.3.7 Ω Λ
The CC Problem These measurements imply a small, positive cosmological constant (small amount of negative energy density) A good understanding of all the contributions to the vacuum energy density is not known. However, any one source (say, from quantum fluctuations in the field of one particle) gives a contribution roughly 120 orders of magnitude larger than the observed value This implies that all these contributions together cancel out to 120 decimal places. It s hard to imagine this happening by accident
High Energy/Short Distance Traditional quantum theories assume pointlike objects as particles From a distance this looks ok, but at very high energies (short distances) we can see that we are containing a finite amount of energy/charge/etc. in an infinitesimal space. This doesn t make sense Most quantum theories avoid this issue: objects aren t really bare particles because they are constantly interacting with virtual particle in the vacuum, etc. These effective objects make sense at low energies. We know the theory still has a problem at high energies, though. Gravity is worse: no way to separate low energy and high energy behavior.
String Theory Naive solution to short distance problem: make fundamental objects not point-like but extended in one dimension: strings Introduces a natural length (the length of the string) Energies, charges, interactions are spread out over this distance, implying there is no ugly singular behavior at high energies
String Theory Unification Old paradigm: patterns of particles at lower energies comes from constituent particles discovered at higher energies molecules atoms baryons, mesons quarks New paradigm: particles have internal structure that leads to patterns, but this structure doesn t come from constituent particles but from being fundamentally extended objects Strings with different shapes give rise to different particles Particle B Particle A
Masses M 1 The mass of a stringy particle comes from its internal energy The more wiggles are on the string, the higher the internal energy This creates a pattern of particles with increasing masses M 2 M 3 M 4 M 1 <M 2 <M 3 <M 4 <M 5 M 5
Spins The spin of a stringy particle comes from it s internal motion Waves travel around the string creating internal angular momentum The more waves, the larger the angular momentum S 1 S 2 S 3 S 4 S 1 <S 2 <S 3 <S 4 <S 5 S 5
Unifying Forces If strings explain the observed forces, we should see photons, gluons, gravitons, etc in the range of particles produced by strings Closed strings: massless, spin 2 particle arises (the graviton) Open strings: massless, spin 1 particle arises (gluons/photons) Most models for string theory assume that the strong and electroweak forces are unified into one force with a large number of massless spin 1 mediators which become the photons, W and Z bosons, and gluons at lower energies Massless, spin 1 open strings would create this family of mediators
Black Holes and Strings? Can string theory explain the information paradox? Assumed solution: GR black hole only an approximation. Actually black holes are complicated objects containing all the information of the objects that fall into them This information would be contained in the strings that make up the black hole The relationship between a stringy black hole and a gravitational black hole has been understood for very special cases, but isn t fully worked out
String Theory and Λ Drawback: string theory only makes sense in a set number of space-time dimensions (10 for string theory, 11 for M-theory) We observe only 4 dimensions. The others must be wrapped up somehow (in circles, spheres, donut shapes, etc) There are lots of different ways these dimensions can be wrapped up, these lead to different 4-d physics (including different values of the vacuum energy) Could one of these explain the observed cosmological constant?
Gravity Summary Gravity is different from the other forces because of the equivalence of gravitational and inertial mass. This leads to the curved space interpretation Black holes are a prediction of this. Hawking radiation from black holes seems to be irreversible, in conflict with QM Gravity interacts with the vacuum energy, Observation shows it is positive and small Gravity quantization breaks down at high energies in a way that is naturally solved by string theory Strings vibrating in a variety of ways give rise to particles of different masses and spins, including gravitons and photons Strings may solve the black hole and cosmological constant problems as well, but this is not yet clear